# Curl reversibility?

1. Mar 26, 2009

### Savant13

I'm working with Maxwell's equations, and I have found the curl of a magnetic field at all points. How can I figure out what the magnetic field is at those points?

2. Mar 26, 2009

### Savant13

Should I be asking the differential equations section?

3. Mar 27, 2009

### clem

It involves solving a partial differential equation with boundary conditions.

4. Mar 27, 2009

### lalbatros

5. Mar 27, 2009

### Savant13

In this case, the magnetic field is being created by an electric dipole consisting of two point particles of equal mass and opposite charge in mutual orbit, not a current, so the Biot-Savart law doesn't apply

6. Mar 27, 2009

### weichi

"There *are* expressions for this. You might be able to find them in something like Boas or Arfken and Weber.

If you are familiar with differential forms, many (most? all?) proofs of the converse of Poincare's lemma also give the expressions that you want. See, e.g, Flanders."

Also, note that the field you are trying to calculate is radiating radiation, so you might want to look at something like Jackson ch 9. Problem 9.1 discusses approaches for solving this type of problem.

7. Mar 27, 2009

### Ben Niehoff

Of course there is a current. Use the continuity equation:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot \vec J = 0$$

Also, as was mentioned, if you are trying to calculate the radiation fields, there is a shortcut. See Jackson.

Also, sometimes (but not always) these things are easier to do in k-space, rather than using curl and grad and such.

8. Mar 27, 2009

### weichi

Ben is of course correct - there *is* a current here. It would be a good exercise to calculate it! Note that this is, in a sense, a "baby" version of the problem you are asking about, but in this case you need to find a vector field who's *divergence* you know.

But once you calculate the current, Biot-Savart isn't going to help you. Do you see why?

9. Mar 27, 2009

### Savant13

Is it because the current is not constant?

10. Mar 27, 2009

### Savant13

Is there any good way to do this that doesn't involve tensors?

11. Mar 28, 2009

### weichi

Yes, exactly! The direction (and location) of the current is always changing. Biot-Savart only applies to steady currents.

Not sure what you mean about tensors, I don't see a use of tensors here.

If you are stuck, why not post what you have so far? Both your solution approach and your result for curl B. (Either on this thread or a new one.) There might be a better way to go about obtaining the solution.

12. Mar 28, 2009

### Savant13

k-Space was mentioned, and I found that k-space requires tensors

13. Mar 29, 2009

### weichi

I don't see why working in k-space would require tensors. I also don't think working in k-space would be helpful for this particular problem, but I admit I haven't given it a great deal of thought.

14. Apr 21, 2009

### Savant13

How would one find the equation based on its divergence?

The divergence is this case is the partial derivative with respect to time of the divergence of the time-varying electric field. So basically what is happening is you take the divergence of the electric field, take the partial derivative of that, and then undo the divergence. I'm not sure if that makes it any easier.

15. Apr 21, 2009

### Savant13

I think I know how I can do this.

Is it possible for a vector field to be perpendicular to its divergence at a point?

16. Apr 21, 2009

### Born2bwire

How can it be perpendicular to its divergence? Divergence results in a scalar.

17. Apr 22, 2009

### Savant13

I'm not sure what I was thinking there, haven't been getting alot of sleep lately.